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Analysis and Applications of Delay Differential Equations inBiology and Medicine

Majid Bani-Yaghoub,Department of Mathematics and Statistics

University of Missouri-Kansas CityKansas City, Missouri 64110-2499, USA

baniyaghoubm@umkc.edu

Abstract

The main purpose of this paper is to provide a summary of the fundamental methods foranalyzing delay differential equations arising in biology and medicine. These methodsare employed to illustrate the effects of time delay on the behavior of solutions, whichinclude destabilization of steady states, periodic and oscillatory solutions, bifurcations,and stability switches. The biological interpretations of delay effects are briefly dis-cussed.

Math. Subj. Classification: 37N25 (Dynamical systems in biology), 37G15(Dynamicalsystems and ergodic theory )

Key Words: Allee Effect; Delay differential Equations; Stability Switch; Bifurcation

1 Introduction

The purpose of this paper is to describe some basic methods that are widely employedin the study of functional differential equations (FDEs) with a special interest on firstand second order Delay Differential Equations (DDEs). Meanwhile, we mention somesignificant outcomes of the analysis of certain DDEs that may be used in the studies ofbiological systems.

There are several great books [42], [3], [27], [26], [5], [50], [62], [45], [36], [46] in thefields of pure and applied mathematics devoted to the qualitative theory of differentialequations with delays. Although some of them might seem very theoretical withoutgiving an outline of the possible applications in biology, they are essential sources andreferences for work on FDEs. Namely, the second version of the book by J.Hale (coau-thored by S.V. Lunel) [42] covers the basic theory of FDEs and also takes into accountmost of the fundamental achievements in the field. This volume refers in particular tothe basic existence theory, properties of the solution map, Liapunov stability theory,stability and boundedness in general linear systems, behaviour near equilibrium andperiodic orbits for autonomous retarded equations, global properties of delay equations

1

and FDEs on manifolds, which makes a great reference for studies related to FDEs.Since these basic developments of FDEs in 1970s, various approaches have been ap-

plied in the study of FDEs. For instance, the fundamental principles underlying theinterrelations between c*-algebra and functional differential objects have been revealedin the book by A. Antonevich [3], where solvability conditions of various FDEs are in-vestigated. In addition, the properties of solutions of FDEs have been examined withrespect to oscillation theory in several study cases [36], [26], [5], [4] in which oscillatoryand nonoscillatory properties of first, second and higher-order delay and neutral delaydifferential equations are addressed.

While there are several contributions to the study of existence of solutions and alsosolvability of FDEs, the foremost concern of many applied mathematicians is the be-haviour of the existing solutions of FDEs. Along with the new methods invented inthe study of FDEs, some major tools such as method of characteristics or method ofLiapunov functionals employed in global and local analysis of ODEs and PDEs havebeen extended to the analysis of FDEs. The present work covers some of the methodsthat have been discussed in the book by Gopalsamy [36] and the book by Kuang [46].

The classical stability theory of ODEs was generalized in the 1970s to investigatethe stability of solutions of retarded functional differential equations (RFDEs) of theform

x(t) = f(t, xt), (1)

where x ∈ C([−σ − τ, σ + A],Rn) with σ ∈ R τ, A ≥ 0) and t ∈ [σ, σ + A], xt ∈ C isdefined as xt(θ) = x(t + θ) for θ ∈ [−τ, 0], f : R× C([−τ, 0],Rn) → R

n .Moreover, suppose that f is uniformly continuous; then the stability of trivial solutionsof system (1) is defined as follows.

Definition 1 Suppose f(t, 0) = 0 for all t ∈ R. The solution x = 0 of equation (1) issaid to be

• (i) stable if for any σ ∈ R, ǫ > 0, there is a δ = δ(ǫ, σ) such that φ ∈ β(0, ǫ)implies xt(σ, φ) ∈ β(0, ǫ) for t ≥ σ, where β(0, ǫ) is an open ball centered at theorigin with radius ǫ.

• (ii) asymptotically stable if it is stable and there is a bo = bo(σ) > 0 such thatφ ∈ β(0, bo) implies x(σ, φ)(t) → 0 as t → ∞.

• (iii) uniformly stable if the number δ in the definition is independent of σ.

• (iv) uniformly asymptotically stable if it is uniformly stable and there is a bo > 0such that, for every η > 0, there is a to(η) such that φ ∈ β(0, bo) implies xt(σ, φ) ∈

2

β(0, η) for t ≥ σ + to(η) for every σ ∈ R.

For a general solution y(t) of system (1), the above-mentioned stability concepts aredefined through the stability for the solution z = 0 of the system

z(t) = f(t, zt + yt)− f(t, yt). (2)

A usual manner in the study of FDEs is to investigate local stability analysis of somespecial solutions (e.g. trivial or constant solutions). For this purpose, the standard ap-proach is to analyze the stability of equations linearized about the special solution.Hence, the stability of the special solution depends on the location of the roots of therelated characteristic equation. Despite the fact that RFDEs share many propertieswith ODEs (and also PDEs), we should emphasize that there are fundamental distinc-tions between the two theories. For instance, the linearized autonomous RFDEs definestrongly continuous semi-groups on the phase space that are not analytic. In fact, thespectra of their generators consist of isolated eigenvalues with finite multiplicities. Toexplain this better, consider the linear delay differential equation

x(t) = −αx(t− τ), (3)

which has a discrete delay term τ . For simplicity let τ = 1. Then equation (3) hasthe solution x : t → eλt if and only if the eigenvalue λ is satisfied in the correspondingcharacteristic equation

λ+ αe−λ = 0. (4)

Nevertheless, only a finite number of eigenvalues may have a non-negative real part.Therefore, the center and unstable manifolds of the trivial solution are finite dimen-sional and the strongly continuous semi-group Γ(t) = exp(αt) related to (3) is notanalytic.

2 Method of Steps to Obtain Numerical Solutions

Several methods have been proposed to solve systems of DDEs. Typical methods forsolving DDEs are the method of characteristics, Laplace transforms and method ofsteps. Moreover, many DDE solvers have been developed since the early 1970s that useRunge-Kutta methods, Hermite interpolation and multistep methods for solving systemsof DDEs. For instance, Matlab solver “dde23” is based on a third-order Runge-Kuttamethod that uses Hermite interpolation of the old and new solution and derivative toobtain an accurate interpolation.Gathering all the methods for solving DDEs numerically or analytically is beyond thescope of the present work. In this section, we provide the method of steps that iscommonly used for solving DDEs subject to an initial history function.

3

Let s > 0, and P (t) be a known function in C([−δ, 0],R). Then the problem is to finda function x(t), t ≥ 0 such that

x(t) = f(t, x(t), x(t− δ)), (5)

subject to the initial condition x(t) = P (t) on [−δ, 0].

• Step 1: If t ∈ [−δ, 0] then x(t) = P (t) = xo(t).

• Step 2: If t ∈ [0, δ] then x(t− δ) = xo(t− δ).So we solve x(t) = f(t, x(t), xo(t− δ)) which gives us the solution x1(t).

• Step 3: If t ∈ [δ, 2δ] then x(t− δ) = x1(t− δ).So we solve x(t) = f(t, x(t), x1(t− δ)) which gives us the solution x2(t).

Hence, for each interval we find a solution for equation (5) and the general solution x(t)includes all solutions xo, x1, ... defined in specific intervals.For instance, let

y(t) = y(t− 1)− y(t), for t > 0, (6)

and for t ∈ [−1, 0],y(t) = (t− 1)2, (7)

(i.e. δ = 1 and initial history function P (t) is given in (7)).Step 1: If t ∈ [−1, 0], then y(t) = (t− 1)2.

Step 2: If t ∈ [0, 1], then t− 1 ∈ [−1, 0], using equation (7) we have

y(t− 1) = (t− 2)2.

Hence the differential equation on [0, 1] is

y(t) = (t− 2)2 − y(t),

which has the solution

y1(t) =1

3(t− 2)3 + Ce−t on [0, 1].

Step 3: For t ∈ [1, 2], t−1 ∈ [0, 1], we solve the differential equation y(t) = y1(t−1)−y(t)and this process continues until the desired time interval is approached.

4

3 Analytical Methods to Study Delay Models

3.1 Method of Reduction to ODEs

In some cases DDEs can be equivalent to systems of ODEs due to the special natureof the kernel functions in the integral terms. The so-called method of the “chain trick”was first introduced by D.M. Fargue [28] in 1973 and has been broadly used since (seefor example [66], [54], [60]). We explain this method for the Lotka-Volterra sytem withdistributed delays which is given by

dxi(t)

dt= xi(t)

(bi +

n∑

j=1

aijxj(t) +

n∑

j=1

bij

∫ t

−∞

fij(t− s)xj(s)ds

), (8)

i = 1, 2, ..., n

where bi, aij, bij(i, j = 1, 2, ..., n) are real constants and fij : [0,∞) 7→ [0,∞) are contin-uous scalar functions known as delay kernels and normalized such that

∫ ∞

0

fij(s)ds = 1; i, j = 1, 2, ..., n. (9)

With specific delay kernels fij , sufficient conditions for global asymptotic stability ofsystem (8)-(9) has been studied by A. Woerz-Busekros [66]. By choosing the kernelfunctions of the form fij(t) = αe−αt, α > 0, system (8) is written as

dxi(t)

dt= xi(t)

(bi +

n∑

j=1

aijxj(t) +

n∑

j=1

βijα

∫ t

−∞

e−α(t−s)xj(s)ds

), (10)

i = 1, 2, ..., n; t > 0,

where bi, aij , βij(i = 1, 2, ..., n) are real constants and α is a positive constant.Define a new set of variables xn+j , j = 1, 2, ..., n so that

xn+j(t) = α

∫ t

−∞

e−α(t−s)xj(s)ds; t > 0. (11)

Using the product rule and the fundamental theorem of calculus we get that

dxn+j(t)

dt= α {xj(t)− xn+j(t)} ; j = 1, 2, ..., n. (12)

Thus, the system (10) of n-integrodifferential equations becomes a system of 2n au-tonomous ordinary differential equations

dxi(t)

dt= xi(t)

(bi +

n∑

j=1

aijxj(t) +

n∑

j=1

βijxn+j

); i = 1, 2, ..., n; (13)

5

dxn+j(t)

dt= α {xj(t)− xn+j(t)} ; j = 1, 2, ...n, .

If x∗ = (x∗1, x

∗2, ..., x

∗n), x

∗1 > 0, i = 1, 2, ..., n is a solution of

n∑

j=1

(aij + βij)x∗j + bi = 0; i = 1, 2, ..., n, (14)

then (x∗1, x

∗2, ..., x

∗n, x

∗n+1, ..., x

∗2n), x

∗n+j = x∗

j , j = 1, 2, ..., n is a componentwise positivesteady state of (13). Asymptotic stability of (x∗

1, ..., x∗2n) for the system of ODEs (13) is

equivalent to that of (x∗1, ..., x

∗n) for DDEs system (10).

3.2 Method of Characteristics

Local stability of a steady state solution of an ODE or PDE system is determined bylinearizing the system at that steady state. The powerful Routh-Hurwitz criterion canbe applied to the corresponding characteristic equations to determine if the real part ofthe roots are negative and if the steady state is stable. The method of characteristicshas been extended to analyze the stability of DDEs. However, there are difficulties inapplying such an extension. As manifested in the following example, in the presenceof delay, the roots of the characteristic equation are functions of delays and hence, itis often a difficult task to apply the method of characteristics and the Routh-Hurwitzcriterion to determine the local stability of steady state solutions. Consider the followingprey-predator system with mutually interfering predators;

dx(t)

dt= x(t)

[γ(1− x(t)

k)− aym(t)

],

dy(t)

dt= bx(t− τ)ym(t− τ))− cy(t), (15)

where a, b, c and k are positive constants and τ ≥ 0 is the discrete delay term, while 0 <m < 1; x(t) and y(t) respectively denote the biomass of prey and predator populations.System (15) has a positive steady state E∗ := (x∗, y∗) satisfying

γ

(1− x∗

k

)= ay∗m,

bx∗(y∗)m−1 = c. (16)

Then by letting x(t) = x∗ +X(t) and y(t) = y∗ + Y (t) and linearizing (15) around E∗,we arrive at

dX(t)

dt= −γ

kx∗X(t)− amx∗(y∗)m−1Y (t), (17)

dY (t)

dt= b(y∗)mX(t− τ) + b(y∗)m−1x∗Y (t− τ)− cY (t), (18)

6

which has the characteristic equation given by

D(λ, τ) =

(λ+ γ

kx∗ amx∗(y∗)m−1

−b(y∗)me−λτ λ+ c− bm(y∗)m−1x∗e−λτ

). (19)

When τ = 0, D(λ, 0) is the usual quadratic equation of the form

D(λ, 0) = λ2 + pλ+ q = 0, (20)

where p = c+ γ

kx∗ − bm(y∗)m−1x∗ and

q = γ

kx∗c− γ

kx∗m(y∗)m−1x∗ + amx∗(y∗)m−1b(y∗)m.

Using (16), we can see that p and q are positive, which implies that (20) has roots withnegative real parts. Then, by the Routh-Hurwitz criterion, E∗ is locally asymptoticallystable.

When τ > 0, due to presence of terms with e−τλ, the characteristic equation D(λ, τ)cannot be explicitly determined and hence, the linear stability analysis of the delay sys-tem (15) via method of characteristics remains vague. In the best case for instance, onecan establish sufficient conditions for the nonexistence of delay induced instability (i.e.conditions that the system (15) remain stable near the steady state E∗ after inducingthe delay τ > 0).

However, one may bypass such difficulty by using the method of Liapunov function-als to obtain sufficient conditions for stability and instability of steady states of DDEs.Moreover, the stability results obtained in this way are often global.

3.3 Method of Liapunov Functionals

Consider the general retarded delay differential equation (RDDE),

x(t) = f(t, xt), (21)

where x ∈ C([−σ−τ, σ+A],Rn) with σ ∈ R; τ, A ≥ 0 and t ∈ [σ, σ+A], xt ∈ C definedas xt(θ) = x(t + θ) for θ ∈ [−τ, 0], f : R× C([−τ, 0],Rn) → R

n is uniformly continuousand f(t, 0) = 0.Let V : R × C → R be a continuous functional and x(σ, φ) be a solution of (21) withinitial value φ at σ (i.e. there is an A > 0 such that x(σ, φ) is a solution of (21) on[σ − τ, σ + A) and xσ(σ, φ) = φ).Denote

v = V (t, φ) = limh→0+1

h[V (t+ h, xt+h(t, φ))− V (t, φ)]. (22)

The following theorem contains uniform (asymptotic) stability and boundedness resultsfor the trivial solution of (21).

7

Theorem 1 Let u(s), v(s), w(s) : R+ → R+ be continuous and nondecreasing; u(s) >

0, v(s) > 0 for s > 0 and u(0) = v(0) = w(0) = 0.The following statements are true:(i) if there is a V : R× C → R such that

u(|φ(0)|) ≤ V (t, φ) ≤ v(|φ|),v(t, φ) ≤ −w(|φ(0)|),

then x = 0 (i.e. the trivial solution of (21)) is uniformly stable.(ii) if in addition to (i) lims→+∞ u(s) = +∞, then the solutions of (21) are uniformlybounded (that is for any α > 0 there is a β = β(α) > 0 such that for all σ ∈ R, φ ∈ C‖φ‖ ≤ α, we have |x(σ, φ)(t)| ≤ β for all t ≥ σ).(iii) if in addition to (i), w(s) > 0 for s > 0, then x = 0 is uniformly asymptoticallystable.

Therefore, the method includes the search of functionals V satisfying the conditions ofTheorem 1 to obtain stability for the trivial solution. For instance, the generalized formof Lotka-Volterra system (8)-(9) is in the following form

dxi(t)

dt= xi(t)

(bi +

n∑

j=1

aijxj(t) +n∑

j=1

bijxj(t− τij) +n∑

j=1

cij

∫ t

−∞

kij(t− s)xj(s)ds

),

(23)t > 0; i = 1, 2, ..., n;

with initial conditions

xi(s) = ϕi(s) ≥ 0; s ∈ (−∞, 0);ϕi(0) > 0; sups≤0

|ϕi(s)| < ∞. (24)

Consider the Liapunov functionals v(t) = v(t, x1(·), ..., xn(·)) defined by

v(t) =

n∑

i=1

|log {xi(t)/x∗i }|+

n∑

j=1

|bij |∫ t

t−τij

∣∣xj(s)− x∗j

∣∣ ds (25)

+n∑

j=1

|cij |∫ ∞

0

|kij(s)|(∫ t

t−s

∣∣xj(u)− x∗j

∣∣ du)ds, for t ≥ 0.

Then using Theorem 1 with a few sufficient conditions, it can be shown that all solutionsof (23) subject to initial conditions (24) satisfy limt→∞ xi(t) = x∗

i ; i = 1, 2., ..., n, wherex∗ = (x∗

1, ..., x∗n) is the positive system (23). Hence, x∗ is a global attractor and using

the above method provides global asymptotic stability of x∗.As stated below, the Liapunov functionals can also give sufficient conditions for the

instability of the solution x = 0 of a RDDE ([46] chapter 2):

8

Theorem 2 Suppose V (φ) is a completely continuous scalar functional on C and thereexists a γ > 0 and an open set U in C such that(i) V (φ) > 0 on U , V (φ) = 0 on the boundary of U ; 0 ∈ cl(U

⋂B(0, γ);

(ii) V (φ) ≤ u(|φ(0)|) on U⋂B(0, γ);

(iii) V (φ) ≥ w(|φ(0)|) for (t, φ) ∈ [0,∞)× U⋂

B(0, γ), whereV (φ) ≡ lim 1

h[V (xt+h(t, φ))− V (φ)] as h → 0+;

and where u(s),w(s) are continuous, positive and increasing for s > 0, u(0) = w(0) = 0.Then the trivial solution of (21) is unstable.

The book by K. Gopalsamy [36] is a collection of different theorems and propositionson the stability of DDEs, many of them use the method of Liapunov functionals toestablish a condition of local and global stability.

In contrast to the favorable outcomes of the Liapunov functionals method men-tioned above, there is a downside in employing this method to real problems arisingfrom mathematical models. It is frequently quite demanding to find a Liapunov func-tional V satisfying the conditions mentioned in Theorem 1 or 2. Similarly, the methodof DDEs stability analysis by employing Razumikin-type Theorems [46] suffers fromdifficulty of finding continuously differentiable functions that satisfy the conditions ofthe theorem. That is the reason many authors resort to the method of characteristic toobtain stability conditions for linear (or linearized) differential equations with discreteor distributed delays. However, in general, determining which of the methods is mostadvantageous over the others depends on the nature of the problem.

3.4 Method of Hopf Bifurcation

The classic Hopf bifurcation theory has been extended to systems of DDEs and alsodelayed PDEs by a number of authors. To explain this better, let us consider the linearsystem,

dx(t)

dt+ ax(t) + bx(t − τ) = 0, (26)

where b > a > 0. Then it has the characteristic equation

λ+ a+ be−λτ = 0. (27)

If λ = µ + iw is a root of (27), then so is λ = µ − iw. A Hopf bifurcation is subjectto existence of a pair of pure imaginary eigenvalues in the case that µ = 0. Then,substituting λ = iw into (27) and solving for τ and w, we get that wo =

√b2 − a2 and

τo = cos−1(−a/b)/wo, where τo represents the Hopf bifurcation value and we have acase that is called “delay induced bifurcation”. In particular, for values of τ near τo,the trivial solution of (26) is asymptotically stable for τ < τo and it loses its stabilitywhen τ > τo. We have a similar situation for all τo + 2kπ, k = 1, 2, .... Thus, for τ = τothe linear variational system (26) has periodic solutions with a period of 2π

wo.

9

A local bifurcation analysis can be conducted by perturbation methods (i.e. letτ = τo+ǫ with 0 < ǫ ≪ 1 and substitute it in the characteristic equation) to demonstratethat small perturbations to the bifurcation value τo may destabilize the periodic solutionsof the linear system (26). The period of the exponentially growing unstable solutionsof (26) can be determined and the existence of nonlinear solutions near the perturbedbifurcation value can be established through the procedure of “two-time asymptotic”(see [58] for more details). The article by Mackey and Milton [55] and also [31] provides agood review of such analysis applied to the study of periodic dynamic diseases. Namely,Cheyne-Stokes respiration (i.e. human respiratory ailment) can be manifested by analteration in the regular breathing pattern that can be presented by a nonlinear delaydifferential equation. Linearizing such equation around its steady state xo gives rise tothe equation (26), where x(t) and τ are respectively the level of arterial carbon dioxideCO2 and the time lag between the oxygenation of the blood in the lungs and monitoringby the chemoreceptors in the brainstem [31]. Ventilation of CO2 in blood is related tox(t) through the Hill function ([59] Chapter 1) where the coefficient b can be written asa product of a constant and evaluated at xo (i.e. b = βv

′

o). In order to be biologicallymeaningful, instead of investigating Hopf bifurcation due to changes of delay τ , considerv

′

o = α to be a Hopf bifurcation value. Then it can be shown that small increases to thevalue α destabilizes the trivial solution of (26) and results in an unstable steady stateand a stable limit cycle with an approximate period of 4τ (see [59] section 1.4) for moredetails). Therefore the period and volume of breathing may dramatically change if thegradient of the ventilation v

′

o becomes too large.Consider now the system,

dx(t)

dt+ ax(t) + bx(t − τ) = f(x(t), x(t− τ)), (28)

as a perturbation of (26) (i.e. f takes small values) and τ as a perturbation of τo. Thensimilar to the classical Hopf Bifurcation Theory [44], [36], the question arises of whetherperiodic solutions of system (26) are stable under such perturbations or not. The per-turbed equation (28) has a periodic solution with a period which is a perturbation ofthat of the linear approximation (26). This has been investigated in several popula-tion dynamic models [25], [42], [22], [64], where in most cases, certain conditions onthe parameter values are required to preserve stability of bifurcating periodic solutionsunder perturbations induced by delay. Chapter 2 of Gopalsamy’s book [36] provides aself-contained demonstration of delay induced bifurcation to periodities of this type.

3.5 Oscillatory and Nonoscillatory Methods

Oscillatory solutions of differential equations with or without delay have been frequentlyencountered in many biological processes described by a mathematical model. Chapters7 - 9 of the book by J.D. Murray [59] provide a thorough background regarding biologicaland physiological oscillators studied via systems of differential equations. An oscillatory

10

solution is generally defined as follows.

Definition 2 A nontrivial solution y is said to be oscillatory if it has arbitrary largezeros for t ≥ to, that is, there exists a sequence of zeros tn (i.e y(tn) = 0) of y such thatlimn→∞ tn = ∞. Otherwise y is said to be nonoscillatory.

For instance, the second order delay differential equation,

y′′

(t) +1

2y

′

(t)− 1

2y(t− π) = 0, for t ≥ 0, (29)

has oscillatory solution y(t) = 1 − sin(t) (i.e. it has an infinite sequence of multiplezeros).

Recent developments in the oscillation theory of DDEs are presented in the bookby R.P. Agarwal et al. [4]. In connection with wave profile equations, the existence ofnonoscillatory solutions of second order DDEs has been examined in Chapter 5 of thisbook (see also [14]). Here, let us begin with the delayed logistic equation (Hutchinson’sequation):

x(t) = γx(t)[1 − x(t− τ)/k], (30)

x(t) = φ(t) on [−τ, 0], (31)

where φ(t) ∈ C([−τ, 0],R) is the initial history function.We may nondimensionalize equation (30) and reduce the number of parameters. Inparticular, let y(t) = −1 + x(t)/k and t = τ t, then (30) can be rewritten as

d

dty(t) = −γτ y(t− 1)[1 + y(t)]. (32)

By dropping the bars from y and t and denoting α = γτ , we have

y(t) = −αy(t− 1)[1 + y(t)], (33)

with a new initial history function φ ∈ C([−1, 0],R).Integrating from (33), it can be observed that

1 + y(t) = (1 + y(to)) exp

{−α

∫ t−1

to−1

y(ξ)dξ

}, (34)

which implies 1 + y(t) > 0 as long as y(ξ) exists on [−1, t− 1]. It can be demonstrated

[67] that the solution y(t) = y(φ)(t) of (33) with initial history function φ is boundedand asymptotically tends to the steady state y(t) ≡ 0 of (33) (i.e. limt→+∞ y(t) = 0) ifα ≤ 3

2, φ(θ) ≥ −1 and φ(0) > −1.

Therefore, the positive steady state x(t) ≡ k of (30) with initial function φ(t) is globallyasymptotically stable for delay τ ≤ 3

2γ.

This is done by considering two cases for y(t). If y(t) is nonoscillatory, then y(t) > 0 or

11

y(t) < 0 for some t ≥ to ≥ 0. Assume first that y(t) > 0 for t ≥ to; then, y(t) < 0 fort ≥ to+1 from (33) (since 1+ y(t) > 0). Hence, y(t) is strictly decreasing for t ≥ to+1.There is a c ≥ 0 such that limt→+∞ y(t) = c. And we must have limt→+∞ y(t) = 0 =−αc(1 + c). Therefore, c = 0. The same conclusion holds for y(t) < 0 for t ≥ to.In addition, if y(t) is oscillatory then the global stability will be derived by using basiccalculus and local maximum minimum properties of y(t).The result can be improved to τ ≤ 37

24γat the cost of considerable elaboration [46].

Nevertheless, the attempt to show α < π2γ

was not successful [34].

Equation (30) for single species growth can be generalized to the following first orderscalar non-autonomous delay equation with negative feedbacks [40].

x(t) =

∫ t

t−τ(t)

n∑

i=1

fi(t, x(s))dµi(t, s), (35)

where τ(t) > 0, fi(t, x) and τ(t) are continuous with respect to their arguments andµi(t, s) is continuous with respect to t, nondecreasing with respect to s and is definedfor all (t, s) ∈ R

2. Then with a few modifications to the previous method, sufficientconditions for global stability of the trivial solution of (35) (with respect to appropriateinitial functions) are established [40].

4 Applications in Biology and Medicine

4.1 Destabilizing Effect of Delay

In this section we discuss the effects of delay on the behavior of solutions for modelsthat have been developed for study in different problems in biology and medicine. Asmentioned in Subsection 3.5, in the study of the delay effects on systems of differentialequations, there are many articles that consider delay as a small perturbation to thesystem. Then the perturbation methods can be used to take advantage of already knownresults in non-delayed systems of ODEs or PDEs. For instance, it can be demonstratedthat the small delays have no influence on the qualitative behaviour of the solution ofthe delayed logistic equation (30); whereas, large delays destabilize its positive steadystate. Such small delays with negligible effects on the behaviour of the solution areoften referred to as harmless delay [32]. One may think that sufficiently small delaysare always harmless and can be ignored in the model analysis but this is not so.

A counter example may be found in the book by Hale ([42] version 1977, page 28)where the trivial solution of

x(t) + 2x(t) = −x(t), (36)

is asymptotically stable, but the trivial solution of

x(t) + 2x(t− τ) = −x(t), (37)

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is unstable to any positive delay τ . Other examples of this type may be found inKolmanovskii and Nosov and the invariant systems studies by Shipanov [61]. Moreover,the destabilizing effect of delay can be seen in general scalar neutral differential equationswith a single delay τ ≥ 0

n∑

k=0

akdk

dtkx(t) +

n∑

k=0

bkdk

dtkx(t− τ) = 0. (38)

Then using the method of characteristic, it can be demonstrated ([46] chapter 3) thatthe trivial solution of equation (38) loses its stability for any τ > 0 when |bn| > 0.

4.2 Oscillation or Nonoscillation Affected by Delay

The effect of delay on the oscillatory and nonoscillatory behaviour of delay differentialequation

dx(t)

dt+ ax(t− τ) = 0, (39)

can be seen in the proposition that follows.

Proposition 1 Let a ∈ (0,∞) and τ ∈ (0,∞). Then all nontrivial solutions of (39)are oscillatory if

aeτ > 1, (40)

and (39) has a nonoscillatory solution if

aeτ ≤ 1. (41)

Such a result is a very special case of a large class of DDEs studied by J. Yan [68].Another example is oscillations in a Lotka-Volterra system that has been well investi-gated by Gopalsamy [35]. So far, we have seen that delay may have an effect on thestability of steady states, asymptotic behaviour of trivial solutions and oscillatory (ornonoscillatory) behaviour of solutions. In the following we will observe that delay maycause phenomena called “stability switches. ”

4.3 Stability Switches

The other phenomena to mention in this section are the stability switches due to changesof delay. Starting with a small delay, as the length of the delay increases, the trivialsolution of DDEs can gain or lose its linear stability. Such phenomena are often calledstability switches. There are plenty of studies providing sufficient conditions for the

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existence or nonexistence of stability switches (see [29] and the references therein).Here, consider the system of DDEs

dx1(t)

dt= r1x1(t)

[k1 + α1x2(t− τ2)

1 + x2(t− τ2)− x1(t)

],

dx2(t)

dt= r2x2(t)

[k2 + α2x1(t− τ1)

1 + x1(t− τ1)− x2(t)

], (42)

where τ1, τ2 ≥ 0 and τ1 + τ2 > 0.The system (42) indicates that the mutualistic or cooperative effects are not realizedinstantaneously but take place with time delays. Linearizing the system around itspositive steady state x∗ and solving the corresponding characteristic equation for λ =α+ iβ, the necessary and sufficient conditions can be found for nonexistence of stabilityswitches. In particular we have the following theorem ([36], Section 3.3).

Theorem 3 Assume that ri, ki, αi > 0 and αi > ki for i = 1, 2; then the positive steadystate x∗ of system (42) is linearly asymptotically stable absolutely in delays (i.e. delayinduced stability switches cannot occur and N∗ is asymptotically stable for all delays).

4.4 Conditions for Delay Independent Stability

Despite the fact that time delays are often thought to have destabilizing or stabilityswitching effects, we may observe cases where local stability of a delay system is notaffected by delay at all. In particular, the following theorem gives sufficient conditionsfor delay independent local stability of a steady state of a delay model.Let x1(t) and x2(t) denote the population densities of two species competing for acommon pool of resources in a temporally uniform environment; let bi and mi (i = 1, 2)denote the respective density dependent birth and death rates (see [57], [18], [33] for anextensive discussion of competition processes). Let τij (i, j = 1, 2) be a set of nonnegativeconstants with τ = max { τij |i, j = 1, 2} so that the population densities are governedby

dx1(t)

dt= b1(x1(t− τ11))−m1(x1(t), x2(t− τ12)),

dx2(t)

dt= b2(x2(t− τ22))−m2(x1(t− τ21), x2(t)), (43)

with initial population size

xi(s) = φi(s) > 0; s ∈ [−τ, 0]; i = 1, 2,

φi ∈ C([−τ, 0],R+), φi 6= 0 on [−τ, 0], i = 1, 2. (44)

The following assumptions on the birth and death rates are made for the system ofDDEs (43):

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(i) bi, mi(i = 1, 2) are continuous with continuous partial derivatives for all xi ≥ 0(i1, 2);also we assume

∂bi∂xi

> 0;∂mi

∂xj

> 0 for xi > 0; i, j = 1, 2; (45)

(ii)bi(0) = 0;mi(0, x2) ≡ 0; (46)

(iii) for some x∗1 > 0, x∗

2 > 0 we have

b1(x∗1)−m1(x

∗1, 0) = 0,

b2(x∗2)−m2(0, x

∗2) = 0; (47)

(iv) there exist positive constants δ1,δ2 such that

b1(δ1)−m1(δ1, x2) < 0,

andb2(δ2)−m2(x1, δ2) < 0, (48)

for x1 ≥ 0, x2 ≥ 0;(v) for the positive steady state (α, β) of (43), we have

b1(α)−m1(α, β) = 0,

b2(β)−m2(α− β) = 0;

(vi)∂m1

∂x1>

∂b1∂x1

+∂m2

∂x1,

∂m2

∂x2>

∂b2∂x2

+∂m1

∂x2.

Theorem 4 Assume that the conditions (i)-(vi) hold for the two species competitionmodel (43) with delays in production and interspecific competitive destruction, then thepositive steady state (α, β) of (43) is (locally) asymptotically stable for all delays τij ≥0; i, j = 1, 2.

4.5 Stability Conditions for DDEs

There are numerous articles employing the above-mentioned methods to establish theconditions for local or global stability of solutions of different delay models. Here, weprovide two of many outcomes of this type available for different DDEs.

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Consider the following second order differential equation with finite number of dis-crete delays

x′′

(t) + a1x′

(t) + aox(t) =n∑

j=1

bjx(t− τj), (49)

where ao, a1 and τj ≥ 0; bj ∈ R for j = 1, ..., n and ao >∑n

j=1 |bj |.Then the following theorem by G. Stepan [63] gives the conditions for uniform asymp-

totic stability of the trivial solution of (49).

Theorem 5 The trivial solution of (49) is uniformly asymptotically stable for all valuesof τj ≥ 0 if either

a1 >

∑n

j=1 |bj |(ao −

∑n

j=1 |bj |) 1

2

, (50)

or

a1 >

n∑

j=1

|bj | τj . (51)

In the case of distributed delay, the equation (49) is changed to

x′′

(t) + a1x′

(t) + aox(t) =

∫ 0

−τ

x(t + θ)dη(θ), (52)

where∫ 0

−τ|dη(θ)| = η < +∞, ao > η and there is a v > 0 such that

∫ 0

−τe−vθ |dη(θ)| <

+∞.Then we have the following theorem:

Theorem 6 The trivial solution of (52) is uniformly asymptotically stable if either

a1 >η

(ao − η)1

2

, (53)

or

a1 >

∫ 0

−τ

|θ| dη(θ). (54)

5 Discussion

Linearization at a steady state is one of the main tools in studying continuous math-ematical delay models representing population or epidemic dynamics [6], [7], [8], [9],[10]. For example, several theorems establish conditions for asymptotic stability of thetrivial solution (i.e. zero solution) of a delay differential equation or a neutral functional

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differential equation (NFDE) through the analysis of their characteristic equations. Nev-ertheless, linearization results often provide information about the behavior of solutionsonly near a steady state. In general, questions such as existence and global stability ofperiodic orbits or oscillations in solutions of a continuous delay model can be answeredby employing asymptotic methods, bifurcation analysis, method of Liapunov function-als and other known methods in global analysis of a system of FDEs or PDEs. Theglobal stability analysis of steady states has been the focus of many researchers examin-ing various delay differential population models. Namely, it is often desirable to obtainsufficient conditions for the global asymptotic stability of the positive steady state of anonlinear differential equation. Furthermore, the global existence of periodic solutionsand also chaotic behavior induced by delay have been investigated in quite a few studies[67], [40], [36], [46], [53]

In the study of qualitative changes to DDEs and NFDEs due to changes of discrete(and also distributed) delays, several authors (e.g. [24], [23], [30]) have encountered thestability switches that may take place for a trivial solution of a non-autonomous DDE.In particular, the stability of the trivial solution will be affected through increases ofthe delay length. Moreover, delay may induce destabilizing effects, oscillatory effects orno qualitative effect at all.

Providing the two cases of oscillatory and nonoscillatory among the necessary trans-formations can be effective in global analysis of DDEs and detecting the sufficient con-ditions of global stability of trivial solutions. Moreover, the method of Liapunov func-tionals and also Razumikhin-type theorems have been frequently applied to the studyof global behavior of solutions. For instance, Razumikhin functions can be used to showthat [39] all positive solutions of (2.30) are attracted by the steady state x∗ when cx∗ > band τ < 1/r (r is a constant). Using the above-mentioned methods, the global and localanalysis of several mathematical models in biology have been investigated in a numberof studies [41], [56], [37], [38], [47], [49], [51], [52].

Outcomes and analysis of delay models [11], [12] are biologically interpreted ac-cording to each research project in which they have been applied. In the following weprovide two well known effects in population biology that have been widely studiedthrough systems of DDEs.

5.1 Allee effect

Global behaviors of the solutions can biologically be interpreted in distinct ways. Theso-called Allee effect [1], [2] relates to a population that has a maximal per capitagrowth rate at intermediate density. When the population becomes too large, the posi-tive feedback effect of aggregation and cooperation may then be dominated by densitydependent stabilizing negative feedback effect due to intraspecific competition arisingfrom excessive crowding and the ensuing shortage of resources.These processes have been studied [15], [17] through global (and local) analysis of sev-eral models such as the following Lotka-Volterra type single species population growth

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[39]x(t) = x(t)[a + bx(t− τ)− cx2(t− τ)], (55)

with x(t) = φ(t) ≥ 0, t ∈ [−τ, 0) and φ(0) > 0, where a, c > 0, b ∈ R and φ ∈C([−τ, 0],R).When τ = 0 and b > 0, the system exhibits the Allee effect. Moreover, equation (55)has a unique positive equilibrium x∗ = 1

2c(b+

√b2 + 4ac).

Then the transformation x(t) = x∗[1 + y(t)] reduces equation (55) to

y(t) = −α(t)y(t− τ), t ≥ 0, (56)

where α(t) = [(2cx∗ − b)x∗ + c(x∗)2y(t− τ)][1 + y(t)].Note that such a transformation is required to make the derivative negative. In thisway it can be shown that the trivial solution is a global attractor of all nonoscillatorysolutions of (55) when 2cx∗ − b > 0. In fact let y(t) > 0 be a nonoscillatory solution,then from (56) we get that y(t) < 0. Since y(t) is nonoscillatory, it is concluded thatlimt→∞ y(t) ≥ 0 which if strictly greater than zero, by letting 2cx∗ − b > 0, we havelimt→∞ y(t) < 0. However, the last inequality implies that limt→∞ y(t) = −∞ which isa contradiction. The case y(t) < 0 is similar to this argument.By using the local maximum or minimum properties the global upper and lower boundsfor oscillatory solutions of (55) are in the form of

e−Mτ ≤ 1 + y(t) ≤ eLx∗τ for t ≥ T,

where y(t) is an oscillatory solution; L,M and T are constants.

5.2 Permanence

While the local and global qualitative properties of a given system of DDEs or delayedPDEs are crucial to mathematically analyze and predict different phenomena in popu-lation biology and epidemiology, an important fundamental property to consider is thepermanence (persistence) of the system in the long run. In particular, the questionis whether the involved populations and/or epidemics will remain permanently in co-existence or one of them will finally survive at the expense of the other’s extinction.Permanence of Lotka-Volterra type systems with delays [21],[65], [19], [48], permanenceof delayed Kolmogorove-type systems [20], [39], and the uniform persistence of func-tional differential equations are the typical works on permanence of DDEs. Note that,the general persistence theory of ODEs (see for example [43]) is a prerequisite to thesestudies.

In conclusion, the present work provides the summary of the basic tools that areused in the studies of delay differential equations. More sophisticated tools [13] [15] [16]have been developed based on the characteristics of each specific model.

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